Work #2: Absolute Value InequalitiesAnswered: 6/7Write The Solution In Interval Notation: 6 ∣ 3 X − 5 ∣ + 8 \textgreater 38 6|3x - 5| + 8 \ \textgreater \ 38 6∣3 X − 5∣ + 8 \textgreater 38

Introduction

Absolute value inequalities are a type of mathematical problem that involves solving equations with absolute values. In this article, we will focus on solving absolute value inequalities in the form of $|ax + b| \geq c$ or $|ax + b| \leq c$. We will use the given problem $6|3x - 5| + 8 \ \textgreater \ 38$ as an example to demonstrate the steps involved in solving absolute value inequalities.

Understanding Absolute Value Inequalities

Absolute value inequalities involve the absolute value of a linear expression. The absolute value of a number is its distance from zero on the number line. For example, the absolute value of $-3$ is $3$, because $-3$ is $3$ units away from zero on the number line.

Step 1: Isolate the Absolute Value Expression

The first step in solving an absolute value inequality is to isolate the absolute value expression. In the given problem, we need to isolate the absolute value expression $|3x - 5|$.

6|3x - 5| + 8 \  \textgreater \  38

Subtract $8$ from both sides of the inequality:

6|3x - 5| \  \textgreater \  30

Divide both sides of the inequality by $6$:

|3x - 5| \  \textgreater \  5

Step 2: Write Two Separate Inequalities

When we have an absolute value inequality in the form of $|ax + b| \geq c$, we can write two separate inequalities:

ax + b \geq c \ \text{or} \ ax + b \leq -c

In our case, we have:

3x - 5 \geq 5 \ \text{or} \ 3x - 5 \leq -5

Step 3: Solve Each Inequality

Now, we need to solve each inequality separately.

Solving the First Inequality

3x - 5 \geq 5

Add $5$ to both sides of the inequality:

3x \geq 10

Divide both sides of the inequality by $3$:

x \geq \frac{10}{3}

Solving the Second Inequality

3x - 5 \leq -5

Add $5$ to both sides of the inequality:

3x \leq 0

Divide both sides of the inequality by $3$:

x \leq 0

Step 4: Combine the Solutions

Now, we need to combine the solutions of both inequalities. We can do this by finding the intersection of the two solution sets.

x \geq \frac{10}{3} \ \text{or} \ x \leq 0

Conclusion

In this article, we have demonstrated the steps involved in solving absolute value inequalities. We used the given problem $6|3x - 5| + 8 \ \textgreater \ 38$ as an example to illustrate the process. By following these steps, you can solve absolute value inequalities and find the solution set in interval notation.

Final Answer

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Introduction

Absolute value inequalities can be a challenging topic for many students. In this article, we will answer some of the most frequently asked questions about absolute value inequalities.

Q: What is an absolute value inequality?

A: An absolute value inequality is a mathematical problem that involves solving equations with absolute values. It is a type of inequality that involves the absolute value of a linear expression.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, you need to follow these steps:

1. Isolate the absolute value expression.
2. Write two separate inequalities.
3. Solve each inequality separately.
4. Combine the solutions.

Q: What is the difference between an absolute value inequality and an absolute value equation?

A: An absolute value equation is a mathematical problem that involves solving equations with absolute values, where the absolute value expression is equal to a constant. An absolute value inequality, on the other hand, is a mathematical problem that involves solving inequalities with absolute values, where the absolute value expression is greater than or less than a constant.

Q: How do I write two separate inequalities when solving an absolute value inequality?

A: When solving an absolute value inequality, you can write two separate inequalities by replacing the absolute value expression with the positive and negative values of the expression.

Q: What is the intersection of two solution sets?

A: The intersection of two solution sets is the set of values that are common to both solution sets. In other words, it is the set of values that satisfy both inequalities.

Q: How do I find the intersection of two solution sets?

A: To find the intersection of two solution sets, you need to find the values that are common to both solution sets. You can do this by finding the overlap between the two solution sets.

Q: What is the union of two solution sets?

A: The union of two solution sets is the set of values that are in either of the two solution sets. In other words, it is the set of values that satisfy at least one of the inequalities.

Q: How do I find the union of two solution sets?

A: To find the union of two solution sets, you need to find the values that are in either of the two solution sets. You can do this by combining the two solution sets.

Q: Can I use a calculator to solve an absolute value inequality?

A: Yes, you can use a calculator to solve an absolute value inequality. However, you need to make sure that the calculator is set to the correct mode and that you are using the correct function.

Q: What are some common mistakes to avoid when solving absolute value inequalities?

A: Some common mistakes to avoid when solving absolute value inequalities include:

• Not isolating the absolute value expression.
• Not writing two separate inequalities.
• Not solving each inequality separately.
• Not combining the solutions correctly.

Conclusion

In this article, we have answered some of the most frequently asked questions about absolute value inequalities. We hope that this article has been helpful in clarifying any confusion you may have had about absolute value inequalities.

Final Tips

• Make sure to follow the steps involved in solving absolute value inequalities.
• Use a calculator to check your answers.
• Practice solving absolute value inequalities to become more confident and proficient.

Additional Resources

• Khan Academy: Absolute Value Inequalities
• Mathway: Absolute Value Inequalities
• Wolfram Alpha: Absolute Value Inequalities

Final Answer

The final answer is $\boxed{(-\infty, 0] \cup [\frac{10}{3}, \infty)}$.